Question

write the function below in the form y = f(u) and u = g(x). then find $\frac{dy}{dx}$ as a function of x. y = $sqrt{3x^{2}+3x + 5}$ write the function in the form y = f(u) and u = g(x). choose the correct answer below. a. y = $sqrt{u}$ and u = 3x^{2}+3x + 5 b. y = 3u^{2}+3u + 5 and u = x c. y = u and u = 3x^{2}+3x + 5 d. y = 3u^{2}+3u + 5 and u = $sqrt{x}$ find $\frac{dy}{dx}$ as a function of x. $\frac{dy}{dx}$ =

Explanation:

Step1: Identify the correct composition

We are given $y = \sqrt{3x^{2}+3x + 5}$. If we let $y=f(u)=\sqrt{u}$ and $u = g(x)=3x^{2}+3x + 5$, we can use the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. The correct answer for the composition is A.

Step2: Find $\frac{dy}{du}$

Differentiate $y = \sqrt{u}=u^{\frac{1}{2}}$ with respect to $u$. Using the power - rule $\frac{d}{du}(u^{n})=nu^{n - 1}$, we have $\frac{dy}{du}=\frac{1}{2}u^{-\frac{1}{2}}=\frac{1}{2\sqrt{u}}$.

Step3: Find $\frac{du}{dx}$

Differentiate $u = 3x^{2}+3x + 5$ with respect to $x$. Using the power - rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, we get $\frac{du}{dx}=6x + 3$.

Step4: Apply the chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}=\frac{1}{2\sqrt{u}}$ and $\frac{du}{dx}=6x + 3$ into the chain - rule formula. Then replace $u$ with $3x^{2}+3x + 5$. So $\frac{dy}{dx}=\frac{6x + 3}{2\sqrt{3x^{2}+3x + 5}}$.

Answer:

A. $y=\sqrt{u}$ and $u = 3x^{2}+3x + 5$
$\frac{dy}{dx}=\frac{6x + 3}{2\sqrt{3x^{2}+3x + 5}}$